on two modifications of er/es/1/m queuing system subject...

Acta Polytechnica Hungarica Vol. 12, No. 2, 2015

– 141 –

On Two Modifications of Er/Es/1/m Queuing

System Subject to Disasters

Michal Dorda, Dusan Teichmann

VSB − Technical University of Ostrava, Faculty of Mechanical Engineering,

Institute of Transport, 17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic,

e-mail: [email protected], [email protected]

Abstract: The paper deals with modelling a finite single-server queuing system with the

server subject to disasters. Inter-arrival times and service times are assumed to follow the

Erlang distribution defined by the shape parameter r or s and the scale parameter rλ or sμ

respectively. We consider two modifications of the model − server failures are supposed to

be operate-independent or operate-dependent. Server failures which have the character of

so-called disasters cause interruption of customer service, emptying the system and balking

incoming customers when the server is down. We assume that random variables relevant to

server failures and repairs are exponentially distributed. The constructed mathematical

model is solved using Matlab to obtain steady-state probabilities which we need to compute

the performance measures. At the conclusion of the paper some results of executed

experiments are shown.

Keywords: Er/Es/1/m; queuing; disasters; method of stages; balking; Matlab

1 Introduction

Queuing theory is a useful tool which enables us to find characteristics of queuing

systems. We can meet queuing systems in many sectors, for example in

informatics, telecommunications, transport or economics. In general, a queuing

system represents a system which serves customers coming into the system. There

are a lot of possible ways to classify queuing systems − for example according to

the type of the input process (Poisson, k Erlang etc.), the service discipline (FCFS,

LCFS, priority queues etc.), the capacity of the queue intended for waiting

customers (systems which do not permit waiting for the service, finite capacity or

the infinity capacity of the queue) and so on. Another criterion of the queuing

system’s classification is whether failures of servers are considered in the model.

For a lot of queuing systems which have already been modelled it is assumed that

no failures of servers can occur. Such queuing systems represent the first group of

queuing systems often called reliable queuing systems. The second group of

queuing systems is represented by the so-called unreliable queuing systems or

queuing systems subject to server breakdowns.

mailto:[email protected]

M. Dorda et al. On Two Modifications of Er/Es/1/m Queuing System Subject to Disasters

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Failures of the server have an obvious impact on the performance measures of the

studied queuing system. It is clear, for example, that the mean number of the

customers in the service for an unreliable queuing system should be less than the

value of the same performance measure for a corresponding queuing system

which is not subject to breakdowns. A lot of policies have been developed; the

policies determine what happens with the customer being served when the server

breaks down. In this paper we consider that failures of the server cause emptying

of the queuing system; meaning that all customers in the system leave it without

being served. Moreover, each customer coming into the system when the server is

broken down is not willing to wait and leaves the system (or is rejected). Such

type of server failures are often called disasters or catastrophes.

Some authors have already modelled queuing systems with server failures having

the character of disasters or catastrophes. Krishna Kumar et al. [1] solved an

M/M/1 queuing system. Catastrophes of the server occur according to the Poisson

process when the server is busy. Whenever a catastrophe occurs the system

empties instantly and all newly arriving customers are lost during the server

repair. Sudhesh [2] studied a similar M/M/1 queue which differs in the fact that

customers entering the system become impatient when the server is down. The

authors of papers [1] and [2] executed transient analysis of the systems, which

means they derived formulas for system state probabilities as functions of the time

t. Yechiali [3] examined an M/M/c queue with random disastrous failures which

cause all present customers to be lost. Customers entering the system during

repairing of the server are considered to be impatient.

Queuing systems in which inter-arrival and service times are considered to follow

the Erlang distribution have already been studied in the past. But in comparison

with queues assuming exponential or general distributed inter-arrival and service

times, the models of queuing systems under the assumption of the Erlang

distribution are not so common, especially in the case of being subject to

breakdowns.

Let us look at some models of reliable queuing systems in which the Erlang

distribution is assumed. Plumchitchom and Thomopoulos [4] studied a single-

server queuing system with Erlang distributed inter-arrival and service times.

Wang and Huang [5] modelled a finite M/Ek/1 queuing system with a removable

server (the server is turned off and turned on depending on the number of

customers in the system). The authors further presented a cost function to

determine the optimal policy. Cost and profit analysis for an M/Ek/1 queuing

system with removable service station was carried out by Mishra and

KumarYadav [6]. Yu et al. [7] developed a model of an M/Ek/1 queuing system

with no damage service interruptions − it is assumed that after the first phase of

service the service process can be interrupted with given probability. Binkowski

and McCarragher [8] employed an Er/Ek/1/N queuing system to model the

operation of a mining stockyard. An optimal management problem of the N-policy

M/Ek/1 queuing system with a removable service station under steady-state


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condition was solved by Pearn and Chang [9]. In the paper [10] written by El-

Paoumy and Ismail a solution of a finite MX/Ek/1/N with bulk arrivals, balking and

reneging is demonstrated. The matrix-geometric solution of the M/Ek/1 queue with

balking and state-dependent service was demonstrated by Yue et al. [11]. Shawky

[12] considered a single-channel service time Erlangian queue with finite source

of customers, one server, finite storage capacity and balking and reneging. Adan et

al. [13] analysed an Ek/Er/c queuing system.

Some authors solved queues subject to failures under the assumption of the Erlang

distribution which was most often applied to model service times. An M/Ek/1

queue with server vacation was studied by Jain and Agrawal [14]. The authors

further assumed that the server can break down when it is busy and the Poisson

arrival rate is state dependent. Wang and Kuo [15] solved an M/Ek/1 machine

repair problem − several identical machines operating under the care of an

unreliable service station. The authors employed matrix geometric method to

derive the steady-state probabilities and developed the steady-state profit function

to find out the optimum number of machines. Kumar et al. [16] considered an

MX/Ek/1 two-phase queuing system with a single removable server and with

gating, server start-up and unpredictable breakdowns.

In this paper we will focus on finite single-server queuing systems with the server

subject to disasters, where inter-arrival times and service times follow Erlang

distribution. Further we will assume that times between failures and times to

repair are exponentially distributed. We employed a “direct” approach to model

the queuing systems consisting of creating a state transition diagram, on the basis

of the diagram we derive a linear equation system describing the system and the

equation system is solved numerically using suitable software; in the paper we

give a hint for solving in Matlab [17]. We hope that this paper makes the solving

of finite Erlang distributed queuing systems possible primarily to non-

mathematicians who are not able to employ the most advanced mathematical

methods used for analytical solving of queuing systems.

The rest of the paper is organized as follows. In Section 2 we will discuss the

necessary assumptions. In Section 3 we will present the mathematical model and

its solving using Matlab. In Section 4 we will present results of some numerical

experiments we did with the proposed model. Please note that the paper is an

extended version of our conference paper presented at the conference

Mathematical Methods in Economics 2012 [18].

2 General Assumptions and Notations

Let us consider a single-server queuing system with a finite capacity equal to m,

where m>1; that means the system has the capacity of m places for customers −

single place in the service and m−1 places intended for the waiting of customers.


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Customers waiting in the queue are served one by one according to the FCFS

service discipline.

Let inter-arrival times follow the Erlang distribution with the shape parameter r≥2

and the scale parameter rλ; therefore the mean inter-arrival time is then equal to

1

r

r. Service times are also Erlang distributed with the shape parameter s≥2

and the scale parameter sμ; thus the mean service time is equal to

1

s

s. We

apply the Erlang distribution because it is able to model time duration of a lot of

practical processes in comparison with the exponential distribution, which is often

used. On the other hand, using the Erlang distribution brings some complications

in modelling the system. However, single-server queues with the Erlang

distribution of inter-arrival times or service times are still solvable using

conventional methods.

Let us assume that the server is successively failure-free (or available) and under

repair. We will assume two modifications. For the first modification we consider

that failures of the server can occur when the server is idle or busy − we say that

server failures are operate-independent. In the case of the second modification we

assume that failures of the server are so-called operate-dependent, which means

the server can break down only when it is servicing a customer. Let us assume for

both modifications that repair of the server is started immediately after

breakdowns, and it immediately starts to operate when repaired.

Now it is necessary to make some assumptions about failure frequency. Due to the

fact that our modifications differ in assumptions about the occurrence of server

failures we have to make different suppositions for individual modifications of the

studied system. Assuming ergodicity (the system has the finite capacity), all of the

possible states of the system can be summarized into three states:

The idle state − no customer is in the system (the system is empty) and

the server is not broken down (is in working condition). Let us denote the

equilibrium probability that the server is idle Pidle.

The busy state − i customers are in the system, where mi ,...,1 ; that

means a customer is in the service and (i−1) customers are waiting in the

queue. The equilibrium probability that the system is found in the busy

state is denoted with Pbusy.

The down state − no customer is in the system (as we are considering

disasters) and the server is broken down and under repair; let us denote

the equilibrium probability of this state Pdown.

As these states are mutually exclusive and exhaustive, the sum of these

probabilities has to be equal to 1:

1 downbusyidle PPP . (1)


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Let us start with the first modification. The server is successively available and

broken down. Let the times the server is available be exponentially distributed

with the parameter η meaning that the mean time the server is available equals the

reciprocal value of the parameter η. Times to repair are exponentially distributed

as well, but with the parameter ζ; the mean time to repair therefore equals to

1. It

is clear that the server’s steady-state availability A (the ratio of time the server is

available in expected value) is equal to:

busyidle PPA

11

1

(2)

and the server’s steady-state unavailability U (the ratio of time the server is broken

down in expected value) is:

downPAU

1 . (3)

And now we have to use similar assumptions about the second modification. Let

us assume that times of overall server working between failures are exponentially

distributed with the parameter η; the mean time of overall server working between

failures is then equal to the reciprocal value of the parameter η. Times to repair are

exponentially distributed as well, but with the parameter ζ; the mean time to repair

is therefore equal to

1 as well.

To express the server’s steady-state availability it is necessary to realize that in

this modification the server failures are not as frequent as in the first modification

for the same value of the parameter η. The value η has to be multiplied by a

coefficient expressed by ratio busy

busyidle

P

PP which takes into account the fact that

the ratio of time the server is idle has no impact on failure frequency. Now, for the

server’s steady-state availability we can write:

busyidle

busyidlebusy

busyidle

busy

busyidle

busy

busyidle

PPPPP

PP

P

PP

P

PP

A

11

1

. (4)

Realizing that busyidledown PPP 1 and substituting it into (4) we can derive an

expected formula in the form:

UPP busydown

. (5)


– 146 –

As far as the behaviour of customers at the moment of the failure is concerned, we

will assume that the system empties after every breakdown of the server; and the

system is empty when the server is down − i.e. failures represent so-called

disasters (or catastrophes) in the system.

Let us mention an example of such queuing system from railway transport.

Marshalling yards represent important nodes on each railway net because they

carry out inbound freight trains classification according to directions of individual

wagons and form new outbound freight trains. Such yards are usually equipped

with corresponding infrastructure consisting of reception sidings, a hump, sorting

sidings and so on. The process of freight trains classification is carried out via the

hump − a train of wagons is shunted from an arrival track over the hump and

individual wagons (or set of wagons) are classified onto sorting sidings according

to their directions.

The hump can be considered to be the server, the inbound freight trains represent

customers and the classification process is their service. However, the

infrastructure belonging to the hump can break down from time to time. For

example, some switches of the ladder below the hump can be broken down so

wagons cannot be classified over the hump. In such cases we must carry out the

classification process in a different way without using the hump.

Another example of such queuing system could be a gas station that is open non-

stop. The station is equipped with one gas pump. Drivers arrive at the gas station

in order to pump and pay. However, the gas station is a technical device so it can

be subject to breakdowns. Because the gas station has only one gas pump, no

driver can be served and therefore drivers do not arrive at the station when the gas

pump is closed (under repair). Also all drivers who are at the station when the gas

pump breaks down leave the station to pump somewhere else.

As we stated before, the failures of the server have an impact on performance

measures, therefore it is important to incorporate the failures in mathematical

models of such queuing systems in order to get non-biased results.

3 Mathematical Model

To model the studied queuing system we applied the method of stages (see for

example Kleinrock [19]). The method utilizes the fact that the Erlang distribution

with the shape parameter r or s and the scale parameter rλ or sμ is a sum of r or s

independent exponential distributions with the same parameter rλ or sμ. The

process of each customer’s arrival consists of r exponential phases and the

customer enters the system (or is rejected when the system is full) after finishing

the last phase. Analogously, the service of each customer consists of s exponential

phases and the customer leaves the system after finishing the last phase. Because

the duration of all phases is exponentially distributed, the queue can be modelled


– 147 –

by a Markov chain. The reason for using the Erlang distribution is that this

distribution is more general than the exponential distribution, which cannot be

used in many practical examples; the Erlang distribution can be used to model

random variables with a coefficient of variation less than 1.

Let us consider a random variable K(t) being the number of the customers found

in the system, a random variable I(t) being the number of finished phases of

customer’s arrival, a random variable J(t) being the number of finished phases of

customer’s service and a random variable F(t) being the number of broken servers

at the time t. On the basis of the assumptions established in Section 2 it is clear

that {K(t), I(t), J(t), F(t)} constitutes a Markov chain with the state space

1,0,0,1,...,0,0,,,, fjrikfjik

0,1,...,0,1,...,0,,...,1,,,, fsjrimkfjik .

Let us note that the first subset contains all the idle and down states and the

second subset the busy states. The system is found in the state (k,i,j,f) at the time t

if K(t)=k, I(t)=i, J(t)=j and F(t)=f; let us denote the corresponding probability

P(k,i,j,f)(t). Complex information about Markov chains can be found for example in

Bolch et al. [20].

Now we would like to set up the mathematical model of the system. At first, let us

establish a group of variables αk, where k=0,1,...,m. The variable αk for k=0,1,...,m

can take its value from the set {0,1}. The variables enable us to create the general

model for both modifications (we can even create other modifications of the

studied system using the variables, for example a modification in which the server

breaks down only when the system is full). The variables will be used as a

multiplier of the failure rate η. For the first modification we have αk=1 for

k=0,1,...,m; that means the server can break down when idle (k=0) or busy

(k=1,...,m). For the second modification we have α0=0 (the server can not break

down when idle) and αk=1 for k=1,...,m (the server can break down when busy).

Now we can illustrate the queuing model graphically on a state transition diagram

(Figure 1). The vertices represent the particular states of the system and the

directed edges indicate the possible transitions with the corresponding rates.

Please note that in Figure 1 only those states are depicted which are necessary for

the formation of an equation system. Due to the fact that some edges lead from

nowhere or point to nowhere in Figure 1, let us comment on such examples:

The all red edges should point to states (0,0,0,1) up to (0,r−1,0,1) (the

down states). But we did not draw all of them to these states because it

would make the diagram more chaotic.

Some green and blue edges are duplicated because the states to which the

edges point or from which they lead are not depicted in Figure 1. For

example, the green edge exiting the state (0,0,0,0) leads to the state

(0,1,0,0), which is not depicted in Figure 1. On the other hand, the green

edge leading to the state (0,i,0,0) exits the state (0,i−1,0,0), which is not


– 148 –

depicted either. The blue edge exiting the state (1,0,0,0) leads to the state

(1,0,1,0) (not depicted) and the blue edge leading to the state (1,0,j,0)

exits the state (1,0,j−1,0) (not depicted too).

Some diagonal green edges lead from nowhere or point to nowhere for

the same reason. For example the green edge exiting the state (1,r−1,0,0)

leads to the state (2,0,0,0) (not depicted) and the green arc leading to the

state (k,0,0,0) exits the state (k−1,r−1,0,0) (not depicted).

Figure 1

The state transition diagram


– 149 –

An original file containing the diagram can be downloaded from a web-page with

the supplementary material − see [21].

Now we apply the global balance principle, which states that for each set of states

X the flow out of the set X is equal to the flow into the set X (see Adan and

Resing [22]). On the basis of the state transition diagram we are able to write the

finite linear equation system of the steady-state balance equations in the form:

1,0,0,00,1,0,10,0,0,00 PPsPr s , (6)

for 1,...,1 ri :

1,0,,00,1,,10,0,1,00,0,,00 isiii PPsPrPr , (7)

for 1,...,1 mk :

0,1,0,10,0,1,10,0,0, skrkkk PsPrPsr , (8)

for 1,...,1,1,...,1 rimk :

0,1,,10,0,1,0,0,, sikikikk PsPrPsr , (9)

for 1,...,1 sj :

0,1,0,10,,0,11 jj PsPsr , (10)

for 1,...,1,1,...,1,,...,1 sjrimk :

0,1,,0,,1,0,,, jikjikjikk PsPrPsr , (11)

for 1,...,1,1,...,2 sjmk :

0,1,0,0,,1,10,,0, jkjrkjkk PsPrPsr , (12)

0,0,1,0,0,1,10,0,0, rmrmmm PrPrPsr , (13)

for 1,...,1 sj :

0,1,0,0,,1,0,,1,10,,0, jmjrmjrmjmm PsPrPrPsr , (14)

for 1,...,1 ri :

0,0,1,0,0,, imimm PrPsr , (15)

m

k

s

jjkkr PPPrPr

1

1

00,,0,0,0,0,001,0,1,01,0,0,0 , (16)

for 1,...,1 ri :

m

k

s

jjikkiii PPPrPr

1

1

00,,,0,0,,001,0,1,01,0,,0 . (17)


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Subtracting the probability on the left side of each equation (6) − (17) we got an

equation system which can be written in the matrix form:

PQ0 T ,

where QT is the transposed infinitesimal generator matrix containing the transition

rates of the Markov process and P is the unknown steady-state probability vector.

Because the matrix QT is singular (the equation set is not linearly independent and

one equation is redundant), it is necessary to incorporate the normalization

condition in the form:

11

1

0

1

00,,,

1

0

1

0,0,,0

m

k

r

i

s

jjik

r

i ffi PP . (18)

3.1 Solving Equation System using Matlab and Performance

Measures

We got the equation system of 12 rsrm linear equations formed by

equations (6) up to (18). The number of the unknown stationary probabilities is

equal to rsrm 2 .

To solve the corresponding equation system we can omit an equation, for example

equation (6). Numerical solving of the system can be performed using Matlab.

However, the applied state description in the form of (k,i,j,f) is four-dimensional

and is very good for the formation of the equation system but is absolutely

unsuitable for the computations in Matlab. Therefore we are obliged to establish

an alternative one-dimensional state description in the following form:

The states (k,i,j,f) for k=1,...,m, i=0,...,r−1, j=0,...,s−1 and f=0 can be

denoted using a single value 11 irjsrk ,

The states (k,i,j,f) for k=0, i=0,...,r−1, j=0 and f=0,1 can be denoted using

a single value 1 irfsrm .

Applying the alternative one-dimensional state description we are able to

transform the equation system in the form we need for using Matlab (we have to

work with matrices). In Matlab we solve the linear system in the form:

PAB ,

where T0;...;1;...;0B , where the value 1 is in the row 1 srm (in the case

that we omit the equation corresponding to the steady-state probability 0,0,0,0P ), A

we get from the matrix QT in which the row 1 srm is substituted by the row

matrix 1;....;1;1 and P is the unknown steady-state probability vector.


– 151 –

After numerical solving of the equation system rewritten in the matrix form we

obtain the stationary probabilities we need in order to compute performance

measures of the studied system.

On the basis of the known stationary probability vector P, the steady-state

probability that the server is idle is equal to:

1

00,0,,0

r

iiidle PP , (19)

the steady-state probability that the server is busy can be expressed by the

formula:

m

k

r

i

s

jjikbusy PP

1

1

0

1

00,,, (20)

and for the equilibrium probability that the server is down it holds:

1

01,0,,0

r

iidown PP . (21)

Now let us consider three performance measures − the mean number of the

customers in the service ES, the mean number of the customers waiting in the

queue EL and the mean number of the broken servers EF. All of them can be

computed according to the formula for the mean value of discrete random

variable, where the random variable 1,0S is the number of customers in the

service, 1,0 mL the number of waiting customers and 1,0F the number

of broken servers.

For the performance measures we can write following formulas:

busyPES , (22)

m

k

r

i

s

jjikPkEL

2

1

0

1

00,,,1 , (23)

downPEF . (24)

The Matlab script (m.file) with defined function enabling computation of

equilibrium probabilities (19), (20) and (21) and performance measures (22), (23)

and (24) is published online − see [21].


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4 Results of Experiments

We performed several experiments with both modifications to demonstrate

solvability of the presented model and to obtain some graphical dependencies.

Applied values of the model parameters are summarized in Table 1.

Table 1

Summary of applied values of the model parameters

Parameter m [-] r [-] rλ

[h-1] s [-]

sμ

[h-1] η [h-1] ζ [h-1]

Value 5 2 18 2 20 0.01 up to 0.1

with step 0.01

0.1 up to 1.0 with

step 0.1

Substituting the values summarized in Table 1 into the model rewritten in Matlab

we are able to compute the steady-state probabilities of the system states and on

the basis of them we get the performance measures ES, EL and EF using formulas

(22), (23) and (24).

The values of the mean number of the customers in the service ES are listed in

Table 2. The upper value corresponds to the queuing system with operate-

independent server failures, the lower value to the queue with operate-dependent

server failures. The data from Table 2 are further shown in Figure 2, the left graph

corresponds to the operate-independent modification and the right graph to the

operate-dependent modification of the studied queuing system.

Comparing the values with each other we can see that increasing value of the

parameter η causes the decrease of the performance measure ES. It is an expected

fact because with increasing value of the parameter η server failures are more

frequent. On the other hand, increasing value of the parameter ζ brings about the

increase of the measure ES. This dependency could also be logically expected

because increasing value of ζ causes shorter times to server repair. Let us note that

the values of ES are greater for the second modification than for the first

modification; it is also logical because it has to hold that the failure frequency of

the operate-dependent modification is lower than the failure frequency of the

operate-independent modification. The failure frequency of the operate-dependent

modification was equal to the failure frequency of the operate-independent

modification only in the case that the Pidle would be equal to zero.

Table 2

The mean number of the customers in the service ES

η / ζ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

0.01 0.769 0.806 0.819 0.825 0.829 0.832 0.834 0.835 0.837 0.837

0.780 0.812 0.823 0.828 0.832 0.834 0.836 0.837 0.838 0.839

0.02 0.704 0.768 0.792 0.804 0.812 0.817 0.821 0.824 0.826 0.828

0.722 0.779 0.799 0.810 0.817 0.821 0.825 0.827 0.829 0.830


– 153 –

0.03 0.648 0.733 0.766 0.784 0.795 0.803 0.808 0.812 0.816 0.818

0.673 0.748 0.777 0.793 0.802 0.809 0.814 0.817 0.820 0.822

0.04 0.601 0.701 0.743 0.765 0.779 0.789 0.796 0.801 0.806 0.809

0.630 0.720 0.757 0.776 0.788 0.797 0.803 0.808 0.811 0.814

0.05 0.560 0.672 0.720 0.747 0.764 0.775 0.784 0.791 0.796 0.800

0.592 0.694 0.737 0.760 0.775 0.785 0.793 0.798 0.803 0.806

0.06 0.524 0.645 0.699 0.729 0.749 0.762 0.772 0.780 0.786 0.791

0.558 0.670 0.718 0.745 0.762 0.774 0.782 0.789 0.794 0.799

0.07 0.493 0.620 0.679 0.713 0.734 0.750 0.761 0.770 0.777 0.782

0.528 0.648 0.700 0.730 0.749 0.763 0.773 0.780 0.786 0.791

0.08 0.464 0.597 0.660 0.697 0.721 0.738 0.750 0.760 0.768 0.774

0.501 0.626 0.684 0.716 0.737 0.752 0.763 0.771 0.778 0.784

0.09 0.439 0.576 0.642 0.681 0.707 0.726 0.739 0.750 0.759 0.766

0.477 0.607 0.667 0.703 0.726 0.742 0.754 0.763 0.770 0.776

0.10 0.417 0.555 0.625 0.666 0.694 0.714 0.729 0.741 0.750 0.757

0.455 0.588 0.652 0.690 0.714 0.732 0.745 0.755 0.763 0.769

Figure 2

The dependence of ES on the parameters η and ζ (first modification on the left, second modification on

the right)

Table 3

The mean number of the customers waiting in the queue EL

η / ζ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

0.01 1.180 1.236 1.256 1.267 1.273 1.277 1.280 1.282 1.284 1.285

1.197 1.246 1.263 1.271 1.277 1.280 1.283 1.285 1.286 1.287

0.02 1.076 1.173 1.210 1.229 1.241 1.249 1.255 1.259 1.263 1.265

1.104 1.190 1.222 1.239 1.249 1.255 1.260 1.264 1.267 1.269

0.03 0.987 1.116 1.167 1.194 1.211 1.222 1.231 1.237 1.242 1.246

1.024 1.139 1.184 1.207 1.222 1.232 1.239 1.244 1.248 1.252

0.04 0.912 1.064 1.126 1.160 1.182 1.196 1.207 1.215 1.222 1.227

0.955 1.092 1.148 1.177 1.196 1.209 1.218 1.225 1.230 1.235


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0.05 0.846 1.015 1.088 1.128 1.154 1.171 1.184 1.194 1.202 1.209

0.894 1.049 1.113 1.149 1.171 1.186 1.197 1.206 1.213 1.218

0.06 0.789 0.971 1.052 1.097 1.127 1.147 1.162 1.174 1.183 1.191

0.840 1.008 1.081 1.121 1.147 1.164 1.177 1.187 1.195 1.202

0.07 0.738 0.930 1.018 1.068 1.101 1.124 1.141 1.154 1.164 1.173

0.791 0.971 1.050 1.095 1.123 1.143 1.158 1.169 1.178 1.186

0.08 0.693 0.891 0.985 1.040 1.076 1.101 1.120 1.135 1.146 1.156

0.748 0.935 1.021 1.069 1.101 1.123 1.139 1.152 1.162 1.170

0.09 0.653 0.856 0.955 1.013 1.052 1.079 1.100 1.116 1.128 1.139

0.709 0.902 0.993 1.045 1.079 1.103 1.121 1.135 1.146 1.155

0.10 0.617 0.823 0.926 0.987 1.029 1.058 1.080 1.097 1.111 1.122

0.673 0.871 0.966 1.022 1.058 1.084 1.103 1.118 1.130 1.140

The values of the mean number of the customers waiting in the service EL are

listed in Table 3 and graphically shown in Figure 3. We can see the same

character of dependencies as in the case of the performance measure ES.

Figure 3

The dependence of EL on the parameters η and ζ (first modification on the left, second modification on

the right)

The values of the mean number of the broken servers EF are listed in Table 4 and

graphically shown in Figure 4. It is logical to expect that the measure EF should

increase with increasing value η and decrease with increasing value of ζ − both

expectations were confirmed by reached results. Furthermore, we can check the

correctness of reached results using formulas (3) and (5).


– 155 –

Figure 4

The dependence of EF on the parameters η and ζ (first modification on the left, second modification on

the right)

Table 4

The mean number of the broken servers EF

η / ζ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

0.01 0.091 0.048 0.032 0.024 0.020 0.016 0.014 0.012 0.011 0.010

0.078 0.041 0.027 0.021 0.017 0.014 0.012 0.010 0.009 0.008

0.02 0.167 0.091 0.063 0.048 0.038 0.032 0.028 0.024 0.022 0.020

0.144 0.078 0.053 0.041 0.033 0.027 0.024 0.021 0.018 0.017

0.03 0.231 0.130 0.091 0.070 0.057 0.048 0.041 0.036 0.032 0.029

0.202 0.112 0.078 0.059 0.048 0.040 0.035 0.031 0.027 0.025

0.04 0.286 0.167 0.118 0.091 0.074 0.062 0.054 0.048 0.043 0.038

0.252 0.144 0.101 0.078 0.063 0.053 0.046 0.040 0.036 0.033

0.05 0.333 0.200 0.143 0.111 0.091 0.077 0.067 0.059 0.053 0.048

0.296 0.174 0.123 0.095 0.078 0.065 0.057 0.050 0.045 0.040

0.06 0.375 0.231 0.167 0.130 0.107 0.091 0.079 0.070 0.063 0.057

0.335 0.201 0.144 0.112 0.091 0.077 0.067 0.059 0.053 0.048

0.07 0.412 0.259 0.189 0.149 0.123 0.104 0.091 0.080 0.072 0.065

0.370 0.227 0.163 0.128 0.105 0.089 0.077 0.068 0.061 0.055

0.08 0.444 0.286 0.211 0.167 0.138 0.118 0.103 0.091 0.082 0.074

0.401 0.251 0.182 0.143 0.118 0.100 0.087 0.077 0.069 0.063

0.09 0.474 0.310 0.231 0.184 0.153 0.130 0.114 0.101 0.091 0.083

0.429 0.273 0.200 0.158 0.131 0.111 0.097 0.086 0.077 0.070

0.10 0.500 0.333 0.250 0.200 0.167 0.143 0.125 0.111 0.100 0.091

0.455 0.294 0.217 0.172 0.143 0.122 0.106 0.094 0.085 0.077

Conclusions

In the paper we discussed two modifications of an Er/Es/1/m queuing system

subject to disasters which cause loss of all customers in the system and balking all

customers incoming to the system while the server is under repair. To solve the


– 156 –

proposed model using Matlab, we developed the one-dimensional system state

description, which enabled us to rewrite the linear equation system into the matrix

form. After numerically solving the equation system we got the steady-state

probabilities we need for computing the performance measures. We focused on

three performance measures – ES, EL and EF.

In the experimental part of the paper we presented the dependencies of the

performance measures on the parameters of η and ζ defining the failure frequency

and the repair rate. Our experiments confirmed that the presented mathematical

model can be successfully applied for solving such queuing system. The

experiments showed the expected dependencies:

The increasing value of η decreases the value of ES and the increasing

value of the parameter ζ increases the value of ES. For the same values of

η and ζ, the value of ES is lower for the operate-independent modification

than for the operate-dependent modification.

The values of EL evince the same character of dependency as the values

of ES.

The value of EF increases with the increasing value of η and decreases

with the increasing value of ζ. The values of EF are greater for the

operate-independent modification than for the operate-dependent

modification.

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